When units in observational studies are clustered in groups, such as students in schools or patients in hospitals, researchers often address confounding by adjusting for cluster-level covariates or cluster membership. In this paper, we develop a unified weighting framework that clarifies how different estimation methods control two distinct sources of imbalance: global balance (differences between treated and control units across clusters) and local balance (differences within clusters). We show that inverse propensity score weighting (IPW) with a random effects propensity score model -- the current standard in the literature -- targets only global balance and constant level shifts across clusters, but imposes no constraints on local balance. We then present two approaches that target both forms of balance. First, hierarchical balancing weights directly control global and local balance through a constrained optimization problem. Second, building on the recently proposed Generalized Mundlak approach, we develop a novel Mundlak balancing weights estimator that adjusts for cluster-level sufficient statistics rather than cluster indicators; this approach can accommodate small clusters where all units are treated or untreated. Critically, these approaches rest on different assumptions: hierarchical balancing weights require only that treatment is ignorable given covariates and cluster membership, while Mundlak methods additionally require an exponential family structure. We then compare these methods in a simulation study and in two applications in education and health services research that exhibit very different cluster structures.

翻译：当观察性研究中的个体被聚类于群体中时，例如学生分布于不同学校或患者分布于不同医院，研究者通常通过调整聚类层面的协变量或聚类归属来处理混杂问题。本文提出一个统一的加权框架，用以阐明不同估计方法如何控制两种不同的失衡来源：全局平衡（处理组与对照组在不同聚类间的差异）和局部平衡（聚类内部的差异）。我们证明，采用随机效应倾向得分模型的逆倾向得分加权法——当前文献中的标准方法——仅针对全局平衡和跨聚类的恒定水平偏移，而对局部平衡未施加任何约束。随后，我们提出两种同时针对两种平衡形式的方法。首先，层次平衡权重通过约束优化问题直接控制全局与局部平衡。其次，基于近期提出的广义蒙德拉克方法，我们开发了一种新颖的蒙德拉克平衡权重估计量，该方法调整的是聚类层面的充分统计量而非聚类指示变量；此方法能够适应所有个体均接受处理或均未接受处理的小规模聚类。关键在于，这些方法基于不同的假设：层次平衡权重仅要求给定协变量和聚类归属时处理可忽略，而蒙德拉克方法额外要求指数族结构。最后，我们通过模拟研究以及在教育和健康服务研究中的两个应用（呈现截然不同的聚类结构）对这些方法进行比较。